Set Species#

class sage.combinat.species.set_species.SetSpecies(min=None, max=None, weight=None)[source]#

Bases: GenericCombinatorialSpecies, UniqueRepresentation

Returns the species of sets.

EXAMPLES:

Sage

sage: E = species.SetSpecies()
sage: E.structures([1,2,3]).list()
[{1, 2, 3}]
sage: E.isotype_generating_series()[0:4]
[1, 1, 1, 1]

sage: S = species.SetSpecies()
sage: c = S.generating_series()[0:3]
sage: S._check()
True
sage: S == loads(dumps(S))
True

Python

>>> from sage.all import *
>>> E = species.SetSpecies()
>>> E.structures([Integer(1),Integer(2),Integer(3)]).list()
[{1, 2, 3}]
>>> E.isotype_generating_series()[Integer(0):Integer(4)]
[1, 1, 1, 1]

>>> S = species.SetSpecies()
>>> c = S.generating_series()[Integer(0):Integer(3)]
>>> S._check()
True
>>> S == loads(dumps(S))
True

class sage.combinat.species.set_species.SetSpeciesStructure(parent, labels, list)[source]#

Bases: GenericSpeciesStructure

automorphism_group()[source]#

Returns the group of permutations whose action on this set leave it fixed. For the species of sets, there is only one isomorphism class, so every permutation is in its automorphism group.

EXAMPLES:

Sage

sage: F = species.SetSpecies()
sage: a = F.structures(["a", "b", "c"]).random_element(); a
{'a', 'b', 'c'}
sage: a.automorphism_group()                                                # needs sage.groups
Symmetric group of order 3! as a permutation group

Python

>>> from sage.all import *
>>> F = species.SetSpecies()
>>> a = F.structures(["a", "b", "c"]).random_element(); a
{'a', 'b', 'c'}
>>> a.automorphism_group()                                                # needs sage.groups
Symmetric group of order 3! as a permutation group

canonical_label()[source]#

EXAMPLES:

Sage

sage: S = species.SetSpecies()
sage: a = S.structures(["a","b","c"]).random_element(); a
{'a', 'b', 'c'}
sage: a.canonical_label()
{'a', 'b', 'c'}

Python

>>> from sage.all import *
>>> S = species.SetSpecies()
>>> a = S.structures(["a","b","c"]).random_element(); a
{'a', 'b', 'c'}
>>> a.canonical_label()
{'a', 'b', 'c'}

transport(perm)[source]#

Returns the transport of this set along the permutation perm.

EXAMPLES:

Sage

sage: F = species.SetSpecies()
sage: a = F.structures(["a", "b", "c"]).random_element(); a
{'a', 'b', 'c'}
sage: p = PermutationGroupElement((1,2))                                    # needs sage.groups
sage: a.transport(p)                                                        # needs sage.groups
{'a', 'b', 'c'}

Python

>>> from sage.all import *
>>> F = species.SetSpecies()
>>> a = F.structures(["a", "b", "c"]).random_element(); a
{'a', 'b', 'c'}
>>> p = PermutationGroupElement((Integer(1),Integer(2)))                                    # needs sage.groups
>>> a.transport(p)                                                        # needs sage.groups
{'a', 'b', 'c'}

sage.combinat.species.set_species.SetSpecies_class[source]#: alias of SetSpecies